The new regulation attempts to create a barrier between religious groups and contraception coverage, through insurers or a third party, that would give religious groups the choice of opting out of such coverage.

Women who chose to use contraception would still have free access to contraception, according to a Newsmax report.

It wasn't immediately clear whether religious groups would accept the new approach.

The U.S. Conference of Catholic Bishops had no immediate reaction, saying they want to study the proposal.

Nebraska Attorney General Jon Bruning said it is "the first critical step to correct a flagrantly unconstitutional mandate."

However, Bruning said the federal government "continues to rely upon accounting gimmicks to address the concerns of religious non-profits that do not qualify for the exemption...and, this rule utterly fails to address the concerns of for-profit corporations with religious objections.”

On the other side of the argument, policy analyst Sarah Lipton-Lubet of the American Civil Liberties Union said the rule appears to meet the ACLU's goal of providing "seamless coverage" of birth control for the affected women.

Houses of worship have always been exempt, but under Obamacare nonprofit religious groups such as universities, charities and hospitals must allow employees to access contraception if the employee choses to do so.

In the latest version of the rule, houses of worship remain exempt. If a nonprofit identifies itself as religious and has private insurance, a private insurer will work directly with women employees to provide coverage for contraception. Many religious nonprofits, however, are self-insured. In those cases, a third party, not the religious employer, would handle the coverage, Newsmax reported.